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Logarithmic Functions

3.3 Differentiation Rules for Exponentials and Logarithms

Differentiation Rules for Exponentials and Logarithms

  • The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
  • All of these rules can be proven via first principles, but it is not required.
  • The basic rules in this section are:

i) If f(x)=a^{x}, \text { then } f^{\prime}(x)=a^{x} \ln a

ii) If f(x)=\log _{a} x, \text { then } f^{\prime}(x)=\frac{1}{x}(\ln a)^{-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

  • Applying the rules to natural exponential and logarithm functions (i.e. a=e), yields:
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3.3 Logarithm Laws [Free]

The Logarithm Function’s Algebraic Properties

  • There are four properties in logarithms to consider:

i) \log_{a}{mn}=\log_{a}{m}=\log_{a}{n}

ii) log_{a}{(\frac{m}{n})}=\log_{a}{m}-\log_{a}{n}

iii) log_{a}{(m^{p})}=p\cdot{\log_{a}{m}}

In particular when p=-1, we have log_{a}{(\frac{1}{m})}=-\log_{a}{m}

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1.4 Logarithmic Functions

The Logarithmic Function

  • The logarithmic function with base is defined as follows:

a^{x} = b is equivalent to log_{a}{b}=x, where a \in R^{+}\setminus \left \{ 1 \right \} .

Note:a \in R^{+}\setminus \left \{ 1 \right \}‘ means that could be any positive number, excluding 1.

  • The expression log_{a}{y}=x is read as: ‘the logarithm of y to the base a is equal to x’.


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